Which of the following equations is an example of inverse variation between the variables [tex]x[/tex] and [tex]y[/tex]?

A. [tex]y=\frac{x}{9}[/tex]
B. [tex]y=9x[/tex]
C. [tex]y=\frac{9}{x}[/tex]
D. [tex]y=x+9[/tex]



Answer :

To determine which of the given equations exemplifies an inverse variation between the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we need to recall the definition of inverse variation. In inverse variation, the product of the two variables is equal to a constant. Mathematically, this relationship can be expressed as:

[tex]\[ y = \frac{k}{x} \][/tex]

where [tex]\( k \)[/tex] is a constant.

Let's evaluate each of the given options one-by-one:

Option A: [tex]\( y = \frac{x}{9} \)[/tex]
- This equation represents a direct variation rather than an inverse variation. Here, [tex]\( y \)[/tex] is directly proportional to [tex]\( x \)[/tex], but scaled by the factor [tex]\(\frac{1}{9}\)[/tex].

Option B: [tex]\( y = 9x \)[/tex]
- This equation also represents a direct variation. In this case, [tex]\( y \)[/tex] is directly proportional to [tex]\( x \)[/tex] with a proportionality factor of 9.

Option C: [tex]\( y = \frac{9}{x} \)[/tex]
- This equation fits the form [tex]\( y = \frac{k}{x} \)[/tex], where [tex]\( k = 9 \)[/tex]. It describes an inverse variation between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Here, the product [tex]\( xy = 9 \)[/tex], which is a constant.

Option D: [tex]\( y = x + 9 \)[/tex]
- This equation represents a linear relationship where [tex]\( y \)[/tex] is shifted by 9 units from [tex]\( x \)[/tex]. It is neither direct nor inverse variation.

Among the given options, option C: [tex]\( y = \frac{9}{x} \)[/tex] is the equation that illustrates inverse variation because it adheres to the form [tex]\( y = \frac{k}{x} \)[/tex].

Thus, the correct answer is:

Option C: [tex]\( y = \frac{9}{x} \)[/tex].